We will review Levy stochastic integrals and Ito’s formula.
Let be a semimartingale, where
is a martingale and
is a process of bounded variation. The problem of stochastic integration is to make sense of objects of the form
If is a Levy process, then there exists
, a Brownian motion
with covariance matrix
in
, and an independent Poisson random measure
on
with Levy measure
, such that for each
, Levy-Ito decomposition has the form of
Given , define
Let (resp.
) be the linear space of all equivalence classes of mappings
which coincide almost everywhere with respect to
, and satisfies the following conditions:
-
is predictable, (see definition here)
-
(resp.
)
(resp.
) is the space of maps
, which is predictable and
(resp.
). Note that
and
are Hilbert spaces, and
and
In this below, we take . We say that an
-valued stochastic process
is a Levy-type stochastic integral if it can be written in the following form for each
,
where for each ,
,
,
, and
is predictable.
is an
-dimensional standard Brownian motion and
is an independent Poisson random measure on
with compensator
and intensity measure
, which we will assume is a Levy measure.
Equivalent notation for Levy-type stochastic integrals of (2) is either
or (to emphasize the domains of integration)
Clearly, is semimartingale, but may not be a Levy process.
Consider a Levy-type stochastic integral of the form (3). Let be the continuous part of
defined by
Theorem 1 (Ito’s theorem) For each
,
, with probability 1,
Another representation of above Ito formula is that, if is of (3), then for each
, with probability 1, we have
Note that a special case of Ito’s formula yields the following classical chain rule for differentiable functions , when the process
is of finite variation:
Here is famous Levy characterization of Brownian motion.
Theorem 2 (Levy’s characterization) Let
be a continuous centered martingale, which is adapted to a given filtration
. If
for each
![]()
where
is a positive definite symmetric matrix, then
is an
-adapted Brownian motion with covariance
.
Proof: Fix and define process
, then by Ito’s formula, we obtain
By taking integral, and expectation
Hence, .
Now, We define the quadratic variation to the more general case of Levy-type stochastic integrals of the form (3). For each
, we define a
matrix-valued adapted process
by the following prescription for its
th entry
Each is almost surely finite, and we have
The term , which sometimes called an Ito correction, arises as a result of the following formal product relations between differentials:
Theorem 3 (Ito’s product formula) If
and
are real-valued Levy-type stochastic integrals of the form (3). Then for all
, we have that
For completeness, we will give another characterization of quadratic variation which is sometimes quite useful. If and
are real-valued Levy-type stochastic integrals of the form (3), then for each
, w.p.1, we have
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